As far as I'm aware, Classics and Archaeology both have a requirement of AAA, so you've definitely got a chance. The college you apply to shouldn't impact your chances of getting in. It's a myth that it's easier to get into the less oversubscribed colleges, since applicants will be pulled to different colleges pre-interview to ensure that no one college is more competitive than the rest.

I got my a level results of AAA and am taking a year or two out.
For my GCSE I got 6A* 7A.

I'm thinking of applying for Classics or Archaeology?

I want to go to a big old college like Balliol or Christ Church.

Truthfully do I stand a chance?

As has been pointed out, you meet the A Level grade entry requirements, and your GCSEs look fine. Classics has an entrance test, and Archaeology requires you to submit written work. Although there is no explicit or definitive guidance about what to do in a gap year, I would suggest that you make sure that you read extensively around your subjects and do some relevant work experience too. That should help with the application and the interview, if you get one.

Otherwise, to reiterate the point, your chance of getting an offer is the same regardless of the college to which you apply.

I got my a level results of AAA and am taking a year or two out.
For my GCSE I got 6A* 7A.

I'm thinking of applying for Classics or Archaeology?

I want to go to a big old college like Balliol or Christ Church.

Truthfully do I stand a chance?

Your grades are good enough - based on that alone you have the same chance as most people applying, which is around 50% for the courses you are applying for (they have two of the highest acceptance rates). Whether you get into your first choice college is something that you can't control - Christ Church for example pools around 2/3 of its applicants to other colleges. If your college is changed you will have to accept the place or reject your offer.

Thank you all for your lovely replies. I'll deffo need to make sure my reasoning for two gap years. Due to personal reasons, I'd never really considered applying to Oxford, but I guess I do actually have a chance then. Need to research courses then, to find what suits me most.

The Oxford website says that the success rate is like 22-23% for Archaeology and 36ish% for Classics ?